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Mathematics

Grade 8

Ratio and proportion

Measurement

Constructions

Module 17

Constructing different angles and triangles

Activity 1

Constructing different angles and triangles

[lo 3.4, 3.5, 4.7]

1. Drawing an angle:Requirements: pencil, ruler, protractor.

1.1 Always begin by drawing a base line.

1.2 Make a mark, e.g. on the left, and position the protractor on the mark.

1.3 Read your protractor from 0°.

1.4 In the case of an angle that is larger than 180°, the relevant angle size must be deducted from 360° before it is drawn. The angle around the outside (the reflex angle) is the angle that you will have to draw.

E.g. 320°: (360° – 320° = 40°). Draw a 40°angle. The reflex angle now represents the 320°.

2. Construct the following angles and name each one:

  • A B ˆ size 12{ { hat {B}}} {} C = 75°

Type of angle:

2.2 C D ˆ size 12{ { hat {D}}} {} E = 135°

Type of angle:

2.3 F G ˆ size 12{ { hat {G}}} {} H = 215°

Type of angle:

3. Constructing a triangle:

Requirements: pencil, ruler, protractor and pair of compasses.

  • Always begin by making a rough sketch.
  • Then use one of the sides of which the length is provided as a base.
  • E.g. construct Δ size 12{Δ} {} ABC with BC = 40 mm, B ˆ size 12{ { hat {B}}} {} = 70° and C ˆ size 12{ { hat {C}}} {} = 50°.

Rough sketch:

  • To measure a lateral length accurately, you must measure the length on you ruler with the help of a pair of compasses. Then the compass point must be positioned on B and the position of C must be indicated with a pencil mark.
  • Construction:

4. Construct each of the following triangles:

4.2 Δ size 12{Δ} {} PQR with QR = 58 mm, P Q ˆ size 12{ { hat {Q}}} {} R = 62° and Q P ˆ size 12{ { hat {P}}} {} R = 69°.

Measure:

  1. PQ = mm
  2. R ˆ size 12{ { hat {R}}} {} =

4.2 Isosceles Δ size 12{Δ} {} ABC with BC = 42 mm, AB = AC and B ˆ size 12{ { hat {B}}} {} = 63°.

Measure:

a) PQ = mm

Activity 2

Bisecting any given line or angle

[lo 3.4, 3.5, 4.7]

  1. Bisecting a given line AB :
  • Measuring line segment AB (e.g. 40 mm).
  • Using a pair of compasses, measure slightly more than half of the line(i.e. ± 22-25 mm).
  • Position the point of the pair of compasses on A and make a pencil stroke below and above the line.
  • Position the point of the compasses on B and draw another pencil stroke above and below the line.
  • Connect the intersections of the pencil strokes.
  • Name the point on line AB , P. P is the centre of line AB .

2. Now try the following:

  • Draw line segment PQ = 70 mm.
  • Bisecting line segment PQ , as in nr. 1 explained.

3. Bisect π ABC :

  • Place the point of the pair of compasses on B .
  • Draw an arc of any size as indicated.
  • Position the point of the compass on the point where the two lines intersect and draw pencil lines inside the angle.
  • Position the point of the compass on the other point of intersection and draw a line inside the angle, so that the two lines intersect.
  • Connect B ˆ size 12{ { hat {B}}} {} (angle B ) with the point where your pencil lines intersect.
  • B ˆ size 12{ { hat {B}}} {} 1 will have the same size as B ˆ size 12{ { hat {B}}} {} 2 . Measure both angles. Are they equal?

4. Try the following:

  • Draw D E ˆ size 12{ { hat {E}}} {} F = 125°.
  • Bisect D E ˆ size 12{ { hat {E}}} {} F .

Activity 3

To construct a line perpendicular from a given point to another line

[lo 3.4, 3.5, 4.7]

1. Construct AD size 12{ ortho } {} BC .

  • Place your compass point on A (you want to draw a perpendicular line on BC from A.)
  • Make an arc on BC .
  • Place the point of your compasses on the one point of intersection between the arc and BC. Draw a line below BC. Place the point of your compasses on the other point of intersection between the arc and BC and draw another line below BC , so that the two lines intersect.
  • Connect A with the intersection of the two drawn lines.
  • Mark the point of intersection D .
  • AD will be perpendicular to BC . ( AD size 12{ ortho } {} BC .)

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Source:  OpenStax, Mathematics grade 8. OpenStax CNX. Sep 11, 2009 Download for free at http://cnx.org/content/col11034/1.1
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